Intuitively, this algebra is an algebra of logical statements in which logically equivalent formulations of the same statement are not distinguished. One can develop intuition for this algrebra by considering a simple case. Suppose our language consists of a number of statement symbols Pi and the connectives∨,∧,¬ and that ⊢
denotes tautologies. Then our algebra consists of statements formed from these connectives with tautologously equivalent satements reckoned as the same element of the algebra. For instance, “¬⁡(P1∧P2)” is
considered the same as “¬⁢P1∨¬⁢P2”. Furthermore, since any statement of propositional calculus may be recast in disjunctive normal form, we may view this particular Lindenbaum-Tarski algebra as a Boolean analogue of polynomials in the Pi’s and their negations.

Lindenbaum-Tarski algebra of a first order langauge

Now, let L be a first order language. As before, we define the equivalence relation ∼ over formulas of L by φ∼ψ if and only if ⊢φ⇔ψ. Let B=L/∼ be the set of equivalence classes. The operations ∨ and ∧ and complementation on B are defined exactly the same way as previously. The resulting algebra is the Lindenbaum-Tarski algebra of the first order language L. It may be shown that

⋁t∈T[φ⁢(t)]

:=[∃x⁢φ⁢(x)]

⋀t∈T[φ⁢(t)]

:=[∀x⁢φ⁢(x)]

where T is the set of all terms in the language L. Basically, these results say that the statement ∃x⁢φ⁢(x) is equivalent to taking the supremum of all statements φ⁢(x) where x ranges over the entire set V of variables. In other words, if one of these statements is true (with truth value 1, as opposed to 0), then ∃x⁢φ⁢(x) is true. The statement ∀x⁢φ⁢(x) can be similarly analyzed.

Remark. It may possible to define the Lindenbaum-Tarski algebra on logical languages other than the classical ones mentioned above, as long as there is a notion of formal proof that can allow the definition of the equivalence relation. For example, one may form the Lindenbaum-Tarski algebra of an intuitionistic propositional language (or predicate language) or a normal modal propositional language. The resulting algebra is a Heyting algebra (or a complete Heyting algebra) for intuitionistic propositional language (or predicate language), or a Boolean algebras with an operator for normal modal propositional languages.